9 QMA Identify natural problems in QMA. Problems known to be in QMA: Local Hamiltonian problem Group Non-membership Approximate shortest vector in a lattice. Is graph non-isomorphism in QMA? Other complexity questions: Is BQP in the polynomial-time hierarchy? Is QIP = PSPACE? Is QIP = EXP?

12 Communication Complexity There are many open questions about quantum communication complexity: Is there a total function for which an exponential savings in communication is possible in the quantum setting? How powerful is prior entanglement for quantum communication protocols? Find a problem for which one-way quantum communication is exponentially more efficient than one-way classical communication.

13 Non-locality and Multiple Provers There are many open questions concerning non-locality. Alice entanglement Bob Referee Referee asks classical questions, Alice and Bob give classical answers we are interested in the possible correlations in their answers.

14 Non-locality and Multiple Provers There are many open questions concerning non-locality. How much classical communication would be needed for classical players Alice and Bob to look quantum to the referee. Cooperative games setting: how much entanglement is needed for Alice and Bob to play optimally? Does parallel repetition work? What is the power of multi-prover quantum interactive proofs.

Factoring by Quantum Computers Ragesh Jaiswal University of California, San Diego A Quantum computer is a device that uses uantum phenomenon to perform a computation. A classical system follows a single

CDMTCS Research Report Series A New Quantum Algorithm for NP-complete Problems Masanori Ohya Igor V. Volovich Science University of Tokyo Steklov Mathematical Institute CDMTCS-194 September 00 Centre for

Quantum Computability and Complexity and the Limits of Quantum Computation Eric Benjamin, Kenny Huang, Amir Kamil, Jimmy Kittiyachavalit University of California, Berkeley December 7, 2003 This paper will

Alternative machine models Computational complexity thesis: All reasonable computer models can simulate one another in polynomial time (i.e. P is robust or machine independent ). But the Turing machine

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Quantum Machine Learning Algorithms: Read the Fine Print Scott Aaronson For twenty years, quantum computing has been catnip to science journalists. Not only would a quantum computer harness the notorious

p. 1/?? Modular arithmetic Much of modern number theory, and many practical problems (including problems in cryptography and computer science), are concerned with modular arithmetic. While this is probably

7-Qubit Quantum Computer Typical Ion Oscillations in a Trap Bits Qubits vs Each qubit can represent both a or at the same time! This phenomenon is known as Superposition. It leads to Quantum Parallelism

Quantum Computing: Lecture Notes Ronald de Wolf Preface These lecture notes were formed in small chunks during my Quantum computing course at the University of Amsterdam, Feb-May 2011, and compiled into

Quantum Computers And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015 1 Includes joint work with Amlan Chakrabarti, U. Calcutta If you were designing Nature, how would

Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986 How to Prove a Theorem So No One Else Can Claim It MANUEL BLUM Goldwasser, Micali, and Rackoff [GMR] define for

How Quantum Can a Computer Be? 1 Elham Kashefi 2 Prisme N 29 September 2014 1 This text is a transcription of the presentation given by Elham Kashefi at the Cournot seminar, «Quantum Turing Testing» in

Introduction to Quantum Computing Frédéric Magniez LIAFA & PCQC, Université Paris Diderot The genesis 2 Copenhagen School (Bohr, Heisenberg, ) - The state of a quantum particule is only fixed after a measurement

1 Digital Signatures A digital signature is a fundamental cryptographic primitive, technologically equivalent to a handwritten signature. In many applications, digital signatures are used as building blocks

1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

EXAM questions for the course TTM4135 - Information Security May 2013 Part 1 This part consists of 5 questions all from one common topic. The number of maximal points for every correctly answered question

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

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The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce

Closed Timelike Curves Make Quantum and Classical Computing Equivalent Scott Aaronson MIT John Watrous University of Waterloo Abstract While closed timelike curves (CTCs) are not known to exist, studying

Quantum and Non-deterministic computers facing NP-completeness Thibaut University of Vienna Dept. of Business Administration Austria Vienna January 29th, 2013 Some pictures come from Wikipedia Introduction

A New Class of Public Key Cryptosystems Constructed Based on Reed-Solomon Codes, K(XII)SEPKC. Along with a presentation of K(XII)SEPKC over the extension field F 2 8 extensively used for present day various

Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design

Introduction to Security Proof of Cryptosystems D. J. Guan November 16, 2007 Abstract Provide proof of security is the most important work in the design of cryptosystems. Problem reduction is a tool to

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CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to